Anillos, repeating that I dont know Irish, it looks as if An Garda Siochána or the Gardai is the proper name of the Irish police force, but that póilineachta is the common noun for police, for example if you were talking about another country's police. So póilineachta is probably the relevant term for this thread. Whether it is dissimilar enough from "police" for OP's purposes I can't say.

and in Irish English usually the Gardai
I agree they often say that, but you often also hear them say "the guards", which is its direct translation into English. Apparently in previous times they were called the Civic Guard, which is very similar in name to what you have in Spain as the Guardia Civil. Btw there is an accent (fada) on the i in An Garda Síochána that you have missed, Vinny, if it bothers you. The o is silent, merely there for the broad/slender spelling rules.
Greek: Αστυνομία (Astynomía, or whatever depending upon you like your upsilons)
Póilíneachta Never heard it . Póilín - na póilíní can be used when talking about generic police but never about our specific Gardaí. Before Independence and the formation of the Garda Síochána the police force in Ireland (Royal Irish Constabulary R.I.C) could have been referred to as the póilíní in Irish. In Irish English sometimes you would hear the pronunciation po-liss instead of police.

#15 - Looks like "Seirbhís Póilíneachta Thuaisceart Éireann" is the official Irish title of the Police Service of Northern Ireland: http://www.psni.police.uk/irish.pdf
"Póilíneachta" would be the genitive of "póilíneacht", which though I haven't heard it before either would logically be an abstract noun derived from "póilín", which means "a policeman". "Póilíní" is the word that's usually used where one might say "police" in English, but it's simply the plural of "póilín".
@Vinny #1: I once heard a shopkeeper in Damascus greet the arrival of a policeman with the words, "Ahleen bil-shurta, ahleen bil-boliis, ahleen bil-hukuumeh!" All apparently synonyms, in this context.